A superset is a set that contains all the elements of another set.
If we have two sets, A and B, we say that A is a superset of B if every element of A is also an element of B. This means if B is a subset of A, then A is the superset of B.
Example
If A = {1, 2, 3, 4, 5, 6, 7} and B = {1, 3, 5}
Since A contains all the elements of B, A is a superset of B.
It is written as A ⊇ B
The line under the symbol ‘⊃’ means that set A may also be equal to set B. In such cases, they will be identical sets. However, if set A is a proper superset of set B (where at least one element of A is not in set B), we remove the line and write A ⊃ B.
Moreover, when a set is not a superset of a given set, we put a slash through the symbol ‘⊃’ and write A ⊅ B (read as A is not a superset of B).
Properties
Since an empty set has no elements, every set is considered a superset of an empty set. For any set A, A ⊃ ɸ
Every set is a superset of itself. For any set A, A ⊃ A
The number of supersets of a set is infinite.
Proper Superset
A proper superset is a set that contains all the elements of another set and at least one additional element.
Set A is a proper superset of set B if every element of B is also an element of A, and A contains at least one element not in B.
Symbol
A proper superset is denoted by the symbol ‘⊃.’ If A is a superset of B and A ≠ B, it is expressed as A ⊃ B, read as ‘proper or strict superset of’ but ‘NOT equal to.’
Examples
If A = {set of quadrilaterals} and B = {set of regular quadrilaterals}
Then, A ⊃ B
Since a set of whole numbers, natural numbers, integers, rational numbers, or irrational numbers are all parts of real numbers, the set of real numbers is the superset of these sets.
ℝ ⊃ ℕ
ℝ ⊃ 𝕎
ℝ ⊃ ℤ
ℝ ⊃ ℚ
ℝ ⊃ ℚ’
Improper Superset
An improper superset is a set that contains all the elements of another set, making it exactly equal to the given set.
An improper superset can sometimes be a superset that is not strictly larger than the subset it contains.
A set A is an improper superset of set B if every element of B is also an element of A, including the possibility that A and B are equal.
Symbol
An improper superset is denoted by the symbol ‘⊇.’ If A is an improper superset of B, it is expressed as A ⊇ B, read simply as ‘superset of,’ where the condition A = B is considered.
Examples
If A = {1, 3, 5} and B = {1, 3, 5}
Then A ⊇ B
Again, if X = {apple, mango, banana} and Y = {mango, banana, apple}
Then X ⊇ Y
Superset vs. Subset
Basis
Superset
Subset
Definition
A set consisting of all elements of the original set.
A set consisting of fewer or the same elements as the original set.
Symbol/Notation
Represented by the notation A ⊇ B (A is a superset of B) or A ⊃ B (A is a proper superset of B).
Represented by the notation A ⊆ B (A is a subset of B) or A ⊂ B(A is a proper subset of B).
Relation to Superset/Subset
The size of the superset is greater than or equal to the size of the subset.
The size of the subset is less than or equal to the size of the superset.
Relation to Empty Set
The empty set (ɸ) is a superset of every set.
The empty set (ɸ) is a subset of every set.
Example
Example:{1, 5, 10} is a superset of {1, 5}.
Example:{1, 5} is a subset of {1, 5, 10}.
Solved Examples
If A = {2, 4, 6, 8, …, 20} and B = {4, 8, 12, 16, 20}, identify the superset and the subset.
Solution:
Given A = {2, 4, 6, 8, …, 20} and B = {4, 8, 12, 16, 20} Since all elements of set B are present in set A, set B is a part of set A. Thus, A is a superset of B, and B is a subset of A.
Determine the superset when two sets are A = {x | x ∈ ℕ} and B = {x | x is an odd number}.
Solution:
Given A = {x | x ∈ ℕ} and B = {x | x is an odd number} A = {1, 2, 3, 4, 5, 6, 7, …} and B = {1, 3, 5, 7, 9, …} Here, set A contains all elements of set B, which means B is a part of A. Thus, A is a superset of B.
